Kirchhoff’s Circuit «Laws» provide a simple approximate
description of how voltage and current behave in electric circuits.
People really like this simple way of looking at things, and they
often take pains to design their circuits so as to minimize the
importance of the various violations of Kirchhoff’s «laws».
However, the direct usefulness of these «laws» is far less than you
might have imagined. The range of useful applicability gets squeezed
from above and from below:

Kirchhoff’s «laws» get squeezed from below by simpler
methods. In particular, the method of series / parallel
reduction applies to a great many (albeit not quite all) practical
circuits. It is simpler, more intuitive, more visual, and generally
easier to apply. In terms of validity, it is no worse – and no
better – than Kirchhoff’s «laws».

Kirchhoff’s «laws» get squeezed from above by more
trustworthy methods. There are plenty of situations where there are
significant, unavoidable violations. You cannot apply these
«laws» safely unless you know when, how, and why they break down.

When in doubt, rely on the Maxwell equations, not Kirchhoff’s
so-called «laws».

Kirchhoff’s Voltage «Law» (KVL) – aka Kirchhoff’s Loop «Law» –
alleges that the voltage around any closed loop is zero. There are
some very serious limitations to the validity of this «law», as
discussed in section 1.5. First, though, let’s figure
out what this «law» tells us, in a best-case scenario.

You might want to review basic concepts of voltage, as set forth in
reference 1. Voltage is defined with respect to a
path. Let’s choose a path, namely a closed loop that goes clockwise,
following the wires and circuit elements. Then KVL alleges that

When applying equation 2a, we need to keep the various
ΔV contributions properly oriented. The rule is to go around
the loop in a particular direction, adding up the voltage
increases. A positive ΔV represents a voltage increase.

Narrowly speaking, that is as far as we can get using KVL alone.
However, we can make additional progress if we bring in Ohm’s
law. For each resistor:

Ohm’s law tells us there is a voltage drop associated with each
resistor. This assumes we have chosen directions consistently, so
that the nominal direction of the current I is the same as the
forward direction along the path we are using to define the voltage.
A voltage drop corresponds to a negative ΔV. For details on
the sign conventions, see note 1.

If you have trouble keeping track of what’s a voltage drop and what’s
a positive ΔV, you can pencil in little positive and negative
signs next to each resistor, as we have done in figure 1. With a little experience, this step becomes
unnecessary.

We can reformulate Kirchhoff’s voltage «law» as follows: Pick any
two points in the circuit (A and B). We suppose there are two or
more ways of getting from A to B, and we call these the legs
of the circuit. Then Kirchhoff’s voltage «law» says that the
voltage increase in each leg is the same.

Let’s apply this to figure 1. We choose point A
to be the southeast corner of the circuit, and point B to be the
northwest corner. There is a black leg (containing the battery) and a
colored leg (containing the two resistors in series). Applying this
to the circuit in figure 1 we find:

We have obtained this directly by comparing legs. (We could also have
obtained it by rewriting equation 4b.)

Real-world electrical engineers are very fond of the leg-by-leg
approach. This is probably because with the loop-by-loop approach,
when there are multiple loops, it can be annoying to keep track of how
the loops mesh together. In contrast, it is straightforward to keep
track of the relationships between all the various legs.

When φ•=0, the leg-by-leg approach is completely
equivalent to the loop approach. Anything you can calculate one way
you can calculate the other way. You should verify this for
yourself.

When φ•≠0, the leg-by-leg approach gives a
more detailed picture of what’s going on. It’s more work, but it’s
more informative.

Also, a reminder: As discussed in section 1.1, there is a
voltage drop as we go along the colored path in the direction of
the current I. By the same token, there is a positive ΔV
if we go along the path in the other direction, from point A to
point B. All this is predicated on I corresponding to positive
current flowing in the direction indicated by the arrowhead on the
diagram.

To say the same thing in words: Given a string of resistors in series,
the voltage drop across one given resistor is to the total voltage
drop as the resistance of the given resistor is to the total
resistance. This is called the voltage-divider rule. It is very
widely useful.

Equation 8 is the rule for resistors in series. It
generalizes to any number of resistors, from one on up. It says that
for resistors in series, just add up all the resistances.

This is important, because in the real world, engineers seldom invoke
KVL directly. Instead, they apply things like equation 8.
This has the advantage of being somewhat simpler and more convenient.
This equation is of course a corollary of KVL, and suffers from all
the limitations as KVL itself, as discussed in
section 1.5.

This is the bread and butter of circuit analysis: Replacing a
complicated circuit by a simpler equivalent circuit.

There are some very serious limitations on the validity of Kirchhoff’s
voltage «law», as we now discuss.

KVL is valid in the low-frequency limit, i.e. valid for DC
circuits.

For AC circuits, KVL is only an approximation. Sometimes it is
a very useful, accurate approximation ... but sometimes it fails
miserably.

It is sometimes said that KVL is exact at all frequencies when
applied to the lumped-circuit model. However, that comes to the
same thing, because for AC circuits the lumped-circuit model is only
an approximation to the real circuit. The failings of the KVL are a
subset of the failings of the lumped-circuit model. See also the
circuit in figure 6. It looks to me like a lumped
circuit in adverse circumstances. It is not consistent with KVL,
and it appears there is no way to remove the inconsistencies.

Here’s a good way of summarizing the situation: KVL is a good
approximation provided there are no significant time-varying
magnetic fields threading the loop – and not otherwise. To
say the same thing in other words: There must not be any significant
parasitic mutual inductances injecting voltage into the circuit.

It is sometimes quite tricky to decide what’s significant and what’s
not.

Very often, engineers take steps to make KVL be a good
approximation, even in situations where it might otherwise have been
much worse. For example, the use of twisted pair might reduce by
several orders of magnitude the amount of flux-dot that couples into
a sensitive part of the circuit.

On the other hand, in the real world, this is not something you can
take for granted. If you want KVL to be a good approximation, you
have to arrange for it by means of suitable engineering. Sometimes
it is not even possible, especially when dealing with high-power,
high-precision, and/or high-frequency circuits.

In situations where KVL does not apply, it’s not the end of the
world. You have to analyze things in more fundamental terms. In
particular, if there’s ever a conflict between Kirchhoff’s «laws»
and the Maxwell equations, go with Maxwell.

We can understand the origins of Kirchhoff’s voltage «law» – and
its limitations – in terms of Faraday’s law of induction, which is
one of the Maxwell equations. It can be written in many forms, of
which the following most directly serves our purposes:

where E is the electric field, B is the magnetic field, S is
some patch of surface, the loop ∂S is the boundary-curve of
that surface, ds is the element of arc length along the boundary,
and dA is the element of area on the surface. This fundamental
law applies to any loop you can think of. Typically the loop is
defined by wires and circuit elements, but this is absolutely not
required. It also applies to any surface you can think of, so long as
∂S is the boundary of S.

If the voltage is a potential, then the loop voltage will be zero, as
called for by Kirchhoff’s «law». However, as discussed in
reference 1, non-potential voltages are often
desirable ... and are often present whether they are desired or not.

Procedures for designing circuitry to minimize the effects of KVL
violations – and for measuring violations when they do occur – are
discussed in reference 2.

Kirchhoff’s Current «Law» – aka Kirchhoff’s Node «Law» – states
that if N current-carrying wires meet at a node, the sum of the
currents must be zero. Inbound currents count as positive, and
outbound currents count as negative. There are very serious
limitations to the validity of this «law», but first let’s see
what this law tells us in a best-case scenario.

We can also interpret this in terms of current in and current out.
The incoming current is I1. The total outgoing current is I2 +
I3. The outgoing current is carried by a parallel combination of
resistors, namely R2 in parallel with R3.

To say the same thing in words: Whenever there is a bunch of resistors
in parallel, the current in one given resistor is to the total current
as the inverse resistance of the given resistor is to the total
inverse resistance. This is called the current-divider rule. It is
very widely useful.

The inverse resistance is called the conductance and is denoted
G. So we can rewrite equation 15 as:

Figure 4 shows the so-called equivalent circuit.
This is identical to Figure 2. It must be emphasized
that this is equivalent only as far as the battery is concerned.
That is to say, it presents an equivalent load on the battery.

This is important, because in the real world, engineers seldom invoke
KCL directly. Instead, they apply things like equation 17.
This has the advantage of being somewhat simpler and more convenient.
This equation is of course a corollary of KCL, and suffers from all
the limitations as KCL itself, as discussed in
section 2.5.

KCL is valid in the low-frequency limit, i.e. valid for DC
circuits, if we exclude situations where Kirchhoff’s third «law»
is violated.

For AC circuits, KCL is only an approximation. Sometimes it is
a very useful, accurate approximation ... but sometimes it fails
miserably.

The fact is, people are accustomed to looking at circuits
where KCL applies, and the question is why. There is no fundamental
physics reason why it should be so. The answer is interesting,
because it has more to do with sociology than physics. The
reason is that people like the simple Kirchhoff way of looking
at things, and designers routinely and habitually take pains to
reduce the significance of KCL violations.

For example, suppose node A carries a high-voltage AC signal, and
suppose node B is a sensitive, high impedance, low voltage
receiver circuit. Any appreciable parasitic capacitance linking
those two nodes would be intolerable. There are many things
we can do to alleviate the problem:

It helps to keep those two nodes far apart ... although
there are limits to how much of this is possible.

It helps a great deal to impose a grounded shield between the
two nodes. This actually increases the amount of parasitic
capacitance, but it reduces the importance. The KCL violation
linking node A to ground is not important, because the current
involved is small compared to other currents flowing through node
A. Meanwhile, the parasitic capacitance connected to node B
is also larger, but it is less important because the driving
voltage is orders of magnitude smaller.

The outer conductor on a coaxial cable is an example of the kind
of shielding we’re talking about. Think about it:

The cable-TV system is built using coax from one end to the
other.

Even at the much lower frequencies used by separated
video, the video input to your TV uses coax.

At yet lower frequencies, the audio inputs to your music
system use coax.

The input to an oscilloscope uses a BNC connector,
because everyone assumes you are going to wire up the
input using coax.

Et cetera.

The point is, without shielding, there would be widespread gross
violations of KCL.

In the real world, KCL cannot be taken for granted. If you want KCL
to be a good approximation, you have to arrange for it by
means of suitable engineering. Sometimes it is not even possible,
especially when dealing with high-power, high-precision, and/or
high-frequency circuits.

In situations where KCL does not apply, it’s not the end of the
world. You just have to analyze things in more fundamental terms.
In particular, if there’s ever a conflict between Kirchhoff’s
«laws» and the Maxwell equations, go with Maxwell.

We can understand the origins of Kirchhoff’s current «law» – and its
limitations – in terms of conservation of charge, which is implied by
the Maxwell equations.

The sum of currents entering the node tells us the rate at which
current is accumulating on the node. In the real world, the node
might well have some finite size and some finite self-capacitance,
in which case it is entirely possible that some charge could
accumulate on the node, in violation of KCL.

In the low-frequency limit, we can make the following argument: We
assume the size of the node is finite, therefore the
self-capacitance of the node is finite. We further assume the
voltage on the node is finite. Combining those two ideas, the
charge on the node must be finite. current is charge per unit time,
and the timescale is very long, so the current must be very small.
Therefore KCL is valid to a good approximation.

In AC circuits, KCL sometimes fails. Significant violations of KCL
can occur even at 60 Hz, which is not a very high frequency. For a
spectacular example of this, see the video in reference 4. If
you consider the helicopter to be a node, unbalanced current is
flowing into the node and back out again, at a frequency of 60 Hz.

On a smaller scale, a very helpful violation of KCL is the basis for
a non-contact voltage detector. The detector is not connected to
the circuit by any conductors. There is also not any need to
connect the detector to ground; all it needs is a large-enough
self-capacitance, like the aformentioned helicopter. The detector
is an isolated node. If there is a nearby wire or other object
carrying an AC voltage, the alternating electric field will give
rise to some charge flowing to and from the detector node, and this
can be measured. See reference 5.

Kirchhoff’s Third law has issues of its own. It is quite
common to have parasitic mutual capacitances between one part of
the circuit and another.

Indeed, in practice, most of the problems with KCL are associated
with parasitic mutual capacitances.

You are encouraged to skip this section, especially on first reading.
The main purpose is to dispel various specific misconceptions. If
you do not suffer from any of these misconceptions, that’s good, and
you should not waste brain cells thinking about them.

By way of background, note that if you go up
stairs in the real world, Δh is positive if h increases
along the path. The same logic applies to electronics: ΔV is
positive if V increases along the path.

Ohm’s law is usually written as an expression for the voltage
drop, as in equation 3. We speak of the IR drop. A drop
corresponds to a negative ΔV if you go along the path
in the direction of the current I.

A lot of people who ought to know better sometimes claim they
have derived KVL as a consequence of conservation of energy. The
derivations are quite ridiculous. You know the result cannot possibly
be true, because energy is always strictly locally conserved,
even in cases where KVL is not valid.

A lot of people who ought to know better sometimes
claim that Kirchhoff’s «laws> are always true. Here’s a better way of
looking at it. It’s a double-or-nothing proposition:

If you encounter a circuit that is working properly, it is
quite likely that Kirchhoff’s «laws» apply to this circuit, to a
reasonable approximation.

If you encounter a circuit that is not
working properly, there is a good chance that Kirchhoff’s «laws»
are being violated.

In particular, when faced with excessive hum or noise or interference,
you should consider all the possible nonidealities that could be
causing the problem. High on the list is the possibility that a KVL
violation could be injecting some unwanted voltage into some leg(s) of
the circuit.

Some engineer designs a circuit. He assumes a priori
that Kirchhoff’s «laws> are always true.

He builds a prototype that doesn’t work properly because of
rampant violations of Kirchhoff’s «laws». He mutters a few
curses, measures the prototype to see where the worst violations are
occuring, and then redesigns the circuit accordingly.

The new design works OK. Now Kirchhoff’s «laws» are valid
a posteriori to a reasonable approxmation.

Note that at the beginning Kirchhoff’s «laws» are assumed valid, and
at the end they are actually valid. However, one must not neglect the
crucial middle stage, where Kirchhoff’s «laws» are not valid. If
the engineer understands why, where, and how much Kirchhoff’s «laws»
are being violated, he will have a much easier time getting past the
middle stage.

KVL and KCL together are called Kirchhoff’s
Circuit «laws». In an appropriately-narrow context, they are often
called simply Kirchhoff’s «laws», but that is risky, because Gustav
Kirchhoff set forth quite a number of laws, on topics including
thermal radiation, spectroscopy, physical chemistry, and circuitry.

Either KVL or KCL separately is sometimes called simply Kirchhoff’s
«law». The meaning is usually obvious from context.

The goal of this section is to understand the voltages and currents in
the mesh circuit shown in figure 6. However,
before we get to that, let’s do a simple warm-up exercise.

Specifically, let’s start by considering the simple circuit shown in
figure 5. There is a square loop made of
resistors. All four resistors have the same value, namely R. There
is an applied magnetic field perpendicular to the plane of the
circuit. The magnetic field is uniform in space. The relevant
component of the magnetic field is steadily increasing. In accordance
with Faraday’s law of induction, this induces an electric field. The
electric field is everywhere parallel to the plane of the mesh.
Roughly speaking, the electric field points in a direction specified
by the three blue arrows surrounding the φ• symbol. This
field is not the gradient of any potential. The total voltage around
the square loop is V0. The voltage drop across each resistor is
V0/4. The current in each resistor is I0/4, where the
“reference current” is I0 := V0/R. This represents a steady
current, an induced current, flowing around the loop. We imagine that
the resistance is small, so the induced current is enormous.

We integrate the magnetic field over the area of the square. This
gives us the magnetic flux, φ. The rate-of-change is constant,
namely φ•. Faraday’s law of induction tells us that the
voltage drop around the square is V0 = φ•.

We are now ready to analyze the mesh circuit shown in figure 6. There are six square cells, all with the same
area. Each cell is bounded by four resistors. Each of the 17
resistors has the same resistance, namely R. Roughly speaking, the
electric field in this case is the superposition of six copies of the
field we saw in figure 5, but in some places the
contributions add or cancel. Finding the details of the electric
field would be more work than necessary, because all we really need to
know is the current in the wires and the voltage measured along
the wires. However, we do know one thing already: The electric field
determines the voltage around each loop, in accordance with the
definition of voltage. For each small square loop, the voltage is
V0.

We consider the following quasi-steady state: The magnetic field is
not steady, but rather the derivative ∂B/∂t is
steady. The electric field is steady. Any initial transients have
died out, so all the voltages and currents in the circuit are steady.
The flux φ is the same from cell to cell and also φ•
is the same from cell to cell. The voltage around each small square
cell is V0 = φ•.

For the six-cell circuit, it is a bit of work to find all the voltages
and currents. On the other hand, given the solution shown in figure 7, it is not hard to verify that the
solution is correct. In the figure, each blue arrowhead represents
I0/7, i.e. one seventh of the reference current. If you multiply
this by R, you find that each blue arrowhead counts for one seventh
of the induced voltage in each cell.

You should check that on the boundary of each cell the net number of
clockwise-pointing blue arrowheads is seven. This means that the
solution upholds Faraday’s law of induction, in agreement with the
amount of φ• specified in figure 6.
You should also verify that the net current into each node is zero, in
accordance with Kirchhoff’s current «law». You should also verify
that everything we have said about the current/voltage relationship is
in agreement with Ohm’s law.

Note that all of these voltages are non-potential voltages. They are
perfectly well defined if you talk about the voltage along a path, but
you get nonsense if you try to talk about “the” voltage at a point.
For example, consider the resistor that proceeds rightward from point
B. I can tell you absolutely that the voltage drop across this
resistor is 4/7ths of V0. I cannot tell you “the” voltage at
either end of this resistor, not even approximately, but I can tell
you the ΔV quite accurately.

Here’s another look at why the voltage must be a functional of the
path, not a function of position:

Consider the clockwise path from point A to point B,
running along the bottom and the left side of the diagram. There is a
drop of 21/7ths i.e. 3 units of voltage along this path.

Consider
the counterclockwise path from point A to point B, running along
the right side and the top of the diagram. There is a gain of 21/7ths
i.e. 3 units of voltage along this path.

Now obviously, you cannot have a voltage at point B that is
simultaneously less than and greater than the voltage at point A.

Note that Kirchhoff’s voltage «law» implies that the voltage is a
potential. KVL cannot possibly give the right answer when applied to
this circuit. Faraday’s law of induction gives the right answer, but
KVL does not.

Also note that you cannot “salvage” KVL by rebuilding the circuit
using different components. In particular, so far as I know, you
cannot set up the pattern of currents and voltages shown in figure 7 using any combination of batteries,
resistors, wires, and other lumped-circuit elements. That’s because
the lumped-circuit model would guarantee that the voltage would be a
potential. It would uphold KVL, whereas the actual circuit does not.
Let’s be clear: KVL says that every time you connect a battery and a
resistor, there is some voltage drop in the resistor along with
(roughly) an equal-sized voltage gain in the battery. The situation
in figure 7 is wildly different: There are
voltage drops (i.e. IR drops) associated with each of the resistors,
but there are no batteries!

Motivation: There are several non-ridiculous ways in which a circuit
like this could come to exist. For one thing, suppose there is a
large time-dependent magnetic field, and you want to screen it, so
that it does not cause horrible interference with all the electronics
in the neighborhood. You might well model the screen in terms of a
mesh. This could be on a smallish scale (such as the ground plane of
a circuit board) or it could be very much larger.

Also, imagine you’re a farmer. There is a lot of metal mesh on the
floor of your dairy barn. There is some old, clunky yet powerful
equipment that puts out a large time-varying magnetic field. You
suspect that this is inducing enormous currents in the mesh. You have
made several valiant attempts to measure the voltages and currents,
but you keep getting inconsistent results ... reproducibly
inconsistent!

As long as you think that KVL applies in this situation, you
will never figure out what’s going on.

As long as you think every voltage is a potential, you will
never figure out what’s going on.

As long as you think is is possible to assign a number to
“the” voltage at a given location, you will never figure out what’s
going on.